Verify Gauss divergence theorem of $F=xi+yj+zk$ over the sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$

zaviknuogg
2022-09-22
Answered

Verify Gauss divergence theorem of $F=xi+yj+zk$ over the sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$

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Ashlynn Delacruz

Answered 2022-09-23
Author has **9** answers

Let ${S}_{a}$ be your sphere, and ${B}_{a}$ the enclosed ball. For each point $\mathbf{r}\in {S}_{a}$ the outwards unit normal n is given by $\mathbf{n}=\frac{\mathbf{r}}{a}$. Furthermore $\mathbf{F}(\mathbf{r})=\mathbf{r}$. Since $\mathbf{r}\cdot \mathbf{r}={a}^{2}$ on ${S}_{a}$ it follows that

${\int}_{{S}_{a}}\mathbf{F}\cdot \mathbf{n}\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\omega ={\int}_{{S}_{a}}\mathbf{r}\cdot \frac{\mathbf{r}}{a}\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\omega =a{\int}_{{S}_{a}}\mathrm{d}\omega =4\pi {a}^{3}\text{}.$

On the other hand, $\mathrm{d}\mathrm{i}\mathrm{v}(\mathbf{F})\equiv 3$, and therefore

${\int}_{{B}_{a}}\mathrm{d}\mathrm{i}\mathrm{v}(\mathbf{F})\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\mathrm{v}\mathrm{o}\mathrm{l}=3\phantom{\rule{thinmathspace}{0ex}}\mathrm{v}\mathrm{o}\mathrm{l}({B}_{a})=4\pi {a}^{3}\text{}.$

${\int}_{{S}_{a}}\mathbf{F}\cdot \mathbf{n}\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\omega ={\int}_{{S}_{a}}\mathbf{r}\cdot \frac{\mathbf{r}}{a}\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\omega =a{\int}_{{S}_{a}}\mathrm{d}\omega =4\pi {a}^{3}\text{}.$

On the other hand, $\mathrm{d}\mathrm{i}\mathrm{v}(\mathbf{F})\equiv 3$, and therefore

${\int}_{{B}_{a}}\mathrm{d}\mathrm{i}\mathrm{v}(\mathbf{F})\phantom{\rule{mediummathspace}{0ex}}\mathrm{d}\mathrm{v}\mathrm{o}\mathrm{l}=3\phantom{\rule{thinmathspace}{0ex}}\mathrm{v}\mathrm{o}\mathrm{l}({B}_{a})=4\pi {a}^{3}\text{}.$

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